Public And Private Identity
An artist website can use the name that best fits the practice. That might be a legal name, artist name, collective name, project name, or pseudonym.
Some artists keep a clear separation between public and personal identity. Others are fully public, or become more public over time. Both approaches are valid.
The useful question is not which choice looks most professional. It is what level of visibility feels sustainable, safe, and true enough to keep using as your art practice evolves.